Fig. 4: Description of the tripartite temporal circuit in terms of time-delocalised subsystems.

a Description of the red circuit fragment in terms of the time-delocalised subsystems A_I, B_I, C_I, Y, Z of the joint system \({T}_{1}{T}_{2}{\bar{T}}_{1}^{{\prime} }{\bar{T}}_{2}^{{\prime} }{Q}_{1}{P}_{O}\), and \({A}_{O},{B}_{O},{C}_{O},\bar{Y},\bar{Z}\) of the joint system \({T}_{1}^{{\prime} }{T}_{2}^{{\prime} }{\bar{T}}_{1}{\bar{T}}_{2}{Q}_{2}^{{\prime} }{F}_{I}\). b Description of the blue circuit fragment in terms of the time-delocalised subsystems A_I, B_I, C_I, Y, Z of the joint system \({T}_{1}{T}_{2}{\bar{T}}_{1}^{{\prime} }{\bar{T}}_{2}^{{\prime} }{Q}_{1}{P}_{O}\), and \({A}_{O},{B}_{O},{C}_{O},\bar{Y},\bar{Z}\) of the joint system \({T}_{1}^{{\prime} }{T}_{2}^{{\prime} }{\bar{T}}_{1}{\bar{T}}_{2}{Q}_{2}^{{\prime} }{F}_{I}\). c The composition of the operations R(U_C) and \({R}^{{\prime} }\) over the systems \(Y,\bar{Y},Z,\bar{Z},{\bar{Q}}_{1},{\bar{Q}}_{2}^{{\prime} }\) gives rise to a cyclic circuit fragment consisting of the operation U_C and the unitary U that defines the process. That is, when evaluating the composition of R(U_C) and R over the wires shown in green (but not over C_I and C_O), we obtain the cyclic circuit in the middle, consisting of the operations U_A, U_B, U_C and U. With respect to the systems \({P}_{O},{A}_{IO}^{{\prime} },{B}_{IO}^{{\prime} },{C}_{IO}^{{\prime} },{F}_{I}\), as well as the time-delocalised systems A_IO, B_IO, C_IO, the circuit therefore consists of U_A, U_B, U_C and U, composed in a cyclic manner as in the process matrix framework.